Subgroups — C₂×C₄²×#170 #13033097

Order 1280.

2046 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#170	C₂	1	6	yes	maximal
2	C₂×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₄²×#170	C₂	1	8	yes	maximal, direct product factor
4	C₂³×#170	C₂²	1	4	yes	—
5	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₅	C₄	1	3	yes	—
7	C₂²×C₄×Dic₅	C₂²	1	3	yes	—
8	C₂×C₄×#170	C₄	1	48	yes	direct product factor
9	C₂×C₄×#170	C₂²	1	24	yes	—
10	C₄²×Dic₅	C₄	1	4	yes	—
11	C₄²×Dic₅	C₂²	1	4	yes	—
12	C₂×C₄²×C₈	—	5	1	no	maximal, Sylow
13	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
14	C₂³×Dic₅	C₂×C₄	1	3	yes	—
15	C₂³×Dic₅	C₂³	1	1	yes	characteristic
16	C₂²×#170	C₂×C₄	1	48	yes	—
17	C₂²×#170	C₂³	1	8	yes	—
18	C₂×C₄²×C₅	C₈	1	8	yes	complemented
19	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
20	C₂×C₄×Dic₅	C₈	1	24	yes	complemented
21	C₂×C₄×Dic₅	C₂×C₄	1	6	yes	—
22	C₂×C₄×Dic₅	C₂×C₄	1	24	yes	—
23	C₂×C₄×Dic₅	C₂×C₄	1	12	yes	—
24	C₂×C₄×Dic₅	C₂³	1	6	yes	—
25	C₄×#170	C₂×C₄	1	96	yes	direct product factor
26	C₂²×C₄×C₈	—	5	6	no	—
27	C₂×C₄³	—	5	1	no	—
28	C₄²×C₈	—	5	8	no	—
29	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅	C₄²	1	12	yes	—
31	C₂²×C₄×C₅	C₂×C₈	1	12	yes	—
32	C₂²×C₄×C₅	C₂×C₈	1	12	yes	—
33	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
34	C₂²×Dic₅	C₄²	1	12	yes	—
35	C₂²×Dic₅	C₂×C₈	1	24	yes	—
36	C₂²×Dic₅	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₅	C₂²×C₄	1	12	yes	—
38	C₂²×Dic₅	C₂²×C₄	1	6	yes	—
39	C₂²×Dic₅	C₂⁴	1	1	yes	characteristic
40	C₂×#170	C₄²	1	64	yes	direct product factor
41	C₂×#170	C₂²×C₄	1	48	yes	—
42	C₄²×C₅	C₂×C₈	1	16	yes	complemented
43	C₄×Dic₅	C₄²	1	24	yes	—
44	C₄×Dic₅	C₂×C₈	1	48	yes	complemented
45	C₄×Dic₅	C₂²×C₄	1	24	yes	—
46	C₂³×C₈	—	5	4	no	—
47	C₂²×C₄²	—	5	3	no	—
48	C₂²×C₄²	—	5	3	no	—
49	C₂²×C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₂×C₄×C₈	—	5	24	no	—
51	C₂×C₄×C₈	—	5	48	no	—
52	C₄³	—	5	4	no	—
53	C₄³	—	5	4	no	—
54	C₂³×C₅	C₂×C₄²	1	6	yes	—
55	C₂³×C₅	C₂²×C₈	1	6	yes	—
56	C₂³×C₅	C₂²×C₈	1	2	yes	—
57	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅	C₄×C₈	1	48	yes	complemented
59	C₂×C₄×C₅	C₂×C₄²	1	12	yes	—
60	C₂×C₄×C₅	C₂²×C₈	1	12	yes	—
61	C₂×C₄×C₅	C₂²×C₈	1	12	yes	—
62	C₂×Dic₅	C₄×C₈	1	48	yes	complemented
63	C₂×Dic₅	C₂×C₄²	1	6	yes	—
64	C₂×Dic₅	C₂×C₄²	1	16	yes	—
65	C₂×Dic₅	C₂×C₄²	1	12	yes	—
66	C₂×Dic₅	C₂²×C₈	1	24	yes	—
67	C₂×Dic₅	C₂³×C₄	1	6	yes	—
68	#170	C₂×C₄²	1	32	yes	direct product factor
69	C₂³×C₄	—	5	1	no	—
70	C₂³×C₄	—	5	3	no	—
71	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
72	C₂²×C₈	—	5	48	no	—
73	C₂²×C₈	—	5	8	no	—
74	C₂×C₄²	—	5	24	no	—
75	C₂×C₄²	—	5	24	no	—
76	C₂×C₄²	—	5	6	no	—
77	C₂×C₄²	—	5	12	no	—
78	C₂×C₄²	—	5	6	no	—
79	C₂×C₄²	#170	1	8	yes	direct product factor
80	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	4	yes	—
81	C₄×C₈	—	5	96	no	—
82	C₂²×C₅	C₄³	1	4	yes	—
83	C₂²×C₅	C₂×C₄×C₈	1	12	yes	—
84	C₂²×C₅	C₂×C₄×C₈	1	12	yes	—
85	C₂²×C₅	C₂²×C₄²	1	3	yes	—
86	C₂²×C₅	C₂³×C₈	1	3	yes	—
87	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₅	C₂×C₄×C₈	1	24	yes	complemented
89	Dic₅	C₄³	1	4	yes	—
90	Dic₅	C₂×C₄×C₈	1	24	yes	complemented
91	Dic₅	C₂²×C₄²	1	4	yes	—
92	C₂⁴	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₂²×C₄	—	5	24	no	—
94	C₂²×C₄	—	5	12	no	—
95	C₂²×C₄	—	5	12	no	—
96	C₂²×C₄	—	5	1	no	—
97	C₂²×C₄	—	5	6	no	—
98	C₂²×C₄	—	5	1	no	—
99	C₂²×C₄	D₅⋊C₈	1	12	yes	—
100	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	12	yes	—
101	C₂²×C₄	C₂×#170	1	12	yes	—
102	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	6	yes	—
103	C₂×C₈	—	5	48	no	—
104	C₂×C₈	—	5	64	no	—
105	C₄²	—	5	48	no	—
106	C₄²	—	5	24	no	—
107	C₄²	—	5	24	no	—
108	C₄²	C₂×#170	1	16	yes	direct product factor
109	C₂×C₅	C₄²×C₈	1	8	yes	complemented
110	C₂×C₅	C₂×C₄³	1	1	yes	characteristic
111	C₂×C₅	C₂²×C₄×C₈	1	3	yes	—
112	C₂×C₅	C₂²×C₄×C₈	1	3	yes	—
113	C₂³	C₂×D₅⋊C₈	1	6	yes	—
114	C₂³	C₂×C₄×AGL(1,𝔽₅)	1	6	yes	—
115	C₂³	C₂²×#170	1	2	yes	—
116	C₂³	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂×C₄	—	5	48	no	—
118	C₂×C₄	—	5	16	no	—
119	C₂×C₄	—	5	24	no	—
120	C₂×C₄	—	5	6	no	—
121	C₂×C₄	—	5	12	no	—
122	C₂×C₄	—	5	6	no	—
123	C₂×C₄	C₄×#170	1	48	yes	direct product factor
124	C₂×C₄	C₂×D₅⋊C₈	1	12	yes	—
125	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	12	yes	—
126	C₂×C₄	C₂²×#170	1	12	yes	—
127	C₈	—	5	32	no	—
128	C₅	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
129	C₂²	C₄×D₅⋊C₈	1	12	yes	—
130	C₂²	C₄²×AGL(1,𝔽₅)	1	4	yes	—
131	C₂²	C₂×C₄×#170	1	12	yes	—
132	C₂²	C₂²×D₅⋊C₈	1	3	yes	—
133	C₂²	C₂²×C₄×AGL(1,𝔽₅)	1	3	yes	—
134	C₂²	C₂³×#170	1	1	yes	characteristic
135	C₄	—	5	24	no	—
136	C₄	—	5	4	no	—
137	C₄	—	5	4	no	—
138	C₄	C₂×C₄×#170	1	24	yes	direct product factor
139	C₂	C₄²×#170	1	8	yes	direct product factor
140	C₂	C₂×C₄×D₅⋊C₈	1	3	yes	—
141	C₂	C₂×C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂	C₂²×C₄×#170	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₄²×C₈
Frattini	C₂³	C₂³×AGL(1,𝔽₅)
Fitting	C₂²×C₄²×C₅	C₄
socle, prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂²×Dic₅	C₂⁴
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₄²×#170	C₂	direct (N × H)
8. C₂×C₄×#170	C₄	direct (N × H)
18. C₂×C₄²×C₅	C₈	semidirect
20. C₂×C₄×Dic₅	C₈	semidirect
25. C₄×#170	C₂×C₄	direct (N × H)
40. C₂×#170	C₄²	direct (N × H)
42. C₄²×C₅	C₂×C₈	semidirect
44. C₄×Dic₅	C₂×C₈	semidirect
58. C₂×C₄×C₅	C₄×C₈	semidirect
62. C₂×Dic₅	C₄×C₈	semidirect
68. #170	C₂×C₄²	direct (N × H)
79. C₂×C₄²	#170	direct (N × H)
88. C₄×C₅	C₂×C₄×C₈	semidirect
90. Dic₅	C₂×C₄×C₈	semidirect
108. C₄²	C₂×#170	direct (N × H)
109. C₂×C₅	C₄²×C₈	semidirect
123. C₂×C₄	C₄×#170	direct (N × H)
128. C₅	C₂×C₄²×C₈	semidirect
138. C₄	C₂×C₄×#170	direct (N × H)
139. C₂	C₄²×#170	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×#170	2
C₂×C₄²×Dic₅	2
C₄²×#170	2
C₂×C₄²×C₈	5

Subgroups — C2×C42×#170 #13033097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×#170 #13033097